It's not currently possible to add extra options to that specific part of the UI. Try voting on this Idea if you want this to be available in the future.
You could try adding a Custom Button in which case a drop-down will appear next to your Edit and Delete buttons with any additional buttons you've added. Alternatively you can use a Publisher Action which will allow you to perform your action from the '+' icon in the bottom right corner on the record detail page.
The publisher actions feature lets you create actions and add them to the Chatter publisher on the home page, on the Chatter tab, in Chatter groups, and on record detail pages. You can also customize the order in which the standard Chatter actions appear, including Post, File, Link, and Poll.